`PFR/HomPFR.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 9 curated AI-written annotations and 485 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Polynomial Freiman-Ruzsa proof paper, arXiv v2

3 declarations, 0 contain sorry, 0 declared as axioms

The homomorphism form of PFR

Here we apply (improved) PFR to show that approximate homomorphisms f from a 2-group to a 2-group are close to actual homomorphisms. Here, approximate is in the sense that f(x+y)-f(x)-f(y) takes few values.

Main results

goursat: A convenient description of the subspaces of $G \times G'$ when $G, G'$ are elementary abelian 2-groups.
homomorphism_pfr : If $f : G → G'$ is a map between finite abelian elementary 2-groups such that $f(x+y)-f(x)-f(y)$ takes at most $K$ values, then there is a homomorphism $\phi: G \to G'$ such that $f(x)-\phi(x)$ takes at most $K^{10}$ values.

Imports (4)

Imported by 3 files

lemma hahn_banach Let $H_0$ be a subgroup of $G$ line 33 source
Source documentation
Let $H_0$ be a subgroup of $G$. Then every homomorphism $\phi: H_0 \to G'$ can be extended to a homomorphism $\tilde \phi: G \to G'$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Module, ZMod, OfNat.ofNat, Nat, AddSubgroup, AddMonoidHom, Subtype, Membership.mem, Exists, Eq
Proof uses: Submodule, OrderIso, AddSubgroup.toZModSubmodule, LinearMap, RingHom.id, AddMonoidHom.toZModLinearMap, LinearMap.comp, Submodule.subtype, LinearMap.exists_extend, Exists.intro, RingHom, id, Eq.mpr, congrArg, Eq.refl
68 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma hahn_banach (H₀ : AddSubgroup G) (φ : H₀ →+ G') : ∃ (φ' : G →+ G'), ∀ x : H₀, φ x = φ' x := by
…
```
lemma goursat Let $H$ be a subgroup of $G \times G'$ line 46 source
Source documentation
Let $H$ be a subgroup of $G \times G'$. Then there exists a subgroup $H_0$ of $G$, a subgroup $H_1$ of $G'$, and a homomorphism $\phi: G \to G'$ such that $$ H := \{ (x, \phi(x) + y): x \in H_0, y \in H_1 \}.$$ In particular, $|H| = |H_0| |H_1|$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Module, ZMod, OfNat.ofNat, Nat, Submodule, Prod, Exists, AddMonoidHom, And, Iff, Membership.mem, Prod.fst, HSub.hSub, Prod.snd, Eq, Nat.card, Subtype, HMul.hMul
Proof uses: LinearEquiv, RingHom.id, LinearMap, LinearEquiv.symm, Prod.mk, HAdd.hAdd, Submodule.exists_equiv_fst_sndModFst, Exists.intro, RingHom, And.intro, Iff.intro, Subtype.mk, Set.mem_of_eq_of_mem, Eq.symm, And.left, Subtype.property, And.right, Eq.mpr, id, congrArg and 6 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
lemma goursat (H : Submodule (ZMod 2) (G × G')) :
    ∃ (H₀ : Submodule (ZMod 2) G) (H₁ : Submodule (ZMod 2) G') (φ : G →+ G'),
      (∀ x : G × G', x ∈ H ↔ (x.1 ∈ H₀ ∧ x.2 - φ x.1 ∈ H₁)) ∧
        Nat.card H = Nat.card H₀ * Nat.card H₁ := by
…
```
theorem homomorphism_pfr Homomorphism PFR line 75 source
Source documentation
Let $f: G \to G'$ be a function, and let $S$ denote the set $$ S := \{ f(x+y)-f(x)-f(y): x,y \in G \}.$$ Then there exists a homomorphism $\phi: G \to G'$ such that $$ |\{f(x) - \phi(x)\}| \leq |S|^{10}. $$
L4DEL explanation
A map between F2-vector spaces that is almost additive differs from a true homomorphism by a set of size at most |S|^10.
1. Statement: if f(x+y) - f(x) - f(y) always lies in a set S, there are a group homomorphism phi and a set T with |T| at most |S|^10 such that f(x) - phi(x) lies in T for all x.
2. Consider the graph A of f inside G x G'. Then A + A sits inside A - ({0} x S), so |A + A| is at most |S| |A|.
3. Apply a PFR covering lemma (better_PFR_conjecture_aux) to get a subgroup H of G x G' and a covering set c of A.
4. Use Goursat's lemma (goursat) to describe H as a graph-like subgroup over phi, then take T as the image of c + ({0} x H1) under the map (x, y) to y - phi(x).
Prerequisites: goursat
See also: PFR_conjecture
Sources and attribution: Pinned Lean source (ce7ac95e620a)
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommGroup, Module, ZMod, OfNat.ofNat, Nat, Finite, Set, Membership.mem, HSub.hSub, HAdd.hAdd, Exists, AddMonoidHom, And, LE.le, Nat.card, HPow.hPow
Proof uses: Fintype, Nonempty, Eq, nonempty_fintype, Nonempty.intro, LT.lt, Nat.card_pos, Prod, Set.graphOn, Set.univ, Real, HMul.hMul, SProd.sprod, Singleton.singleton, Iff.mp, Set.mem_add, Eq.mpr, id, congrArg, propext and 223 more
267 instances, coercions and auxiliary constants left out.

Tactic trace: 107 steps
Open to load the checked before-and-after goal states.
```
theorem homomorphism_pfr (f : G → G') (S : Set G') (hS : ∀ x y : G, f (x+y) - (f x) - (f y) ∈ S) :
    ∃ (φ : G →+ G') (T : Set G'), Nat.card T ≤ Nat.card S ^ 10 ∧ ∀ x : G, (f x) - (φ x) ∈ T := by
…
```